Marvin Minsky: What makes mathematics hard to learn? (2008)

More likely she's played 2048 a couple times. Obligatory xkcd:

https://xkcd.com/1344/

Likewise, I identify with you on the main points. I liked maths as a schooler, did well in it. As did yourself, I had to memorise formulae for exam and was only allowed a 'basic' calculator. I have maths to thank for allowing me to get the necessary school marks to secure my further studies (which are unrelated to maths). But math's honestly just felt like imitating a machine or a robot. Completely mechanical, it just became a competition of making sure you could make the fewest mistakes in the upcoming exam versus your colleagues.

A majority of it is memorisation, which makes it both very easy in some respects, and hard in others. I still feel in the end it is a bit useless to learn it in the way we do now, seeing as computers can do it in a fraction of a second.

In the decimal system it is just as easy to associate 5+5=10.

5xN is easier to remember than 16xN on account of 5 being the smaller number.

2^n is logarithmic whereas 5*n is linear. Arguably, logarithms are not too complicated, even if linear seems to be a degree easier, seeing that the decimal system is also logarithmic as that's a denser representation.

edit: how to enter an aterisk as the multiplication operator sign

There are four other essays in this series.

Effects Of Grade-Based Segregation http://web.media.mit.edu/~minsky/OLPC-2.html

Role Models, Mentors, and Imprimers and Thinking http://web.media.mit.edu/~minsky/OLPC-3.html

Questioning “General” Education http://web.media.mit.edu/~minsky/OLPC-4.html

Education and Psychology http://web.media.mit.edu/~minsky/OLPC-5.html

Mathematics is hard to learn because the way classes and performance testing is done, there is often only one answer and until you get to the answer the question is not complete. the end state is binary - either the student was successful in it, or the student was not.

On the other hand, in a subject like history a student can make good ground towards an answer by laying down fundamental facts and then arguing and reasoning back and forth. Even with minimal facts the student may use some approximate recollection of events and still be able to build an argument. The answer may not be complete, and it may not even be good, but the final outcome has the appearance of being complete, and is often somewhat far from the starting point of nothing.

Humanities subjects use assessments that provide intermediate rewards for a student. Mathematics offer no such respite. That is what I believe makes it intimidating. Even worse, some questions have intermediate solutions look extremely hairy, leading the student to believe that he has gone further away from the solution. In such cases, the intermediate step has punished rather than reward!

Very often suggestions for improving mathematics pedagogy is to make use of "open-ended" questions that admit multiple solutions (as Marvin did here). I believe most of us who make these suggestions understand the effect that traditional maths questions have on a learner. These suggested questions are just like those in history or literature that allow students to have a decent go at it and still make progress.

I don't know if I truly believe in the theory (hypothesis) that memorization makes it extremely boring. Until historical accounts are utilized in some argument (during a performance assessment), these are just empty facts to be memorized. And given the wide range of questions that can be asked in a history exam, the effort involved in memory work in history class seem to be more onerous and less rewarding. Yet I have seen students who are gifted in remembering in detail historical events, capable of building logical structures and narrative flow, and they struggle with maths. So clearly they must have committed a decent amount of effort to memorization - but not in maths but other subjects instead.

In any case, at the higher levels, rote work in mathematics is extremely important: it is very difficult to pursue an advanced class in probability theory without the basic tools picked up in real analysis, for example (that means all your theorems relating to sequences and continuity and functional analysis and so on). That usually involves some amount of "practice", thinking and solving problem sets before moving on to more advanced topics.